Variables

Excerpts

• Variables

  Thus the “field” of the relation for $x$ is restricted to numbers less than $1$, and similarly for the “field” open to $y$.

• Variables

  The ideas of *any* and of *some* are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic.

• Variables

  Thus, as here used, *any* implies *some* and *some* does not exclude *any*.

• Variables

  When we have asked the question implied in the statement of the equation $x + 2 = 3$, $x$ is called the unknown.

• Variables

  One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations.

• Variables

  Then the law, known as Boyle’s law, expressing the relation between $p$ and $v$ as both vary, is that the product $pv$ is constant, always supposing that the temperature does not alter.

• Variables

  In other words the really fundamental idea is that of the pair of *variables* satisfying the relation $pv = 1$.

• Variables

  The Romans would have stated the number of the year in which this is written in the form MDCCCCX., whereas we write it 1910, thus leaving the letters for the other usage.

Equations

• Variables

  x + 2 = 2 + x

  For any number x, adding 2 gives the same result as adding x to 2, so the order of addition does not matter.

• Variables

  x + y = y + x

  For any two numbers x and y, x + y equals y + x (the commutative law of addition, stated for any pair).

• Variables

  x + 2 = 3

  For some number x, x + 2 equals 3; the book notes that the only such number is 1, so this is an equation whose unknown is determined.

• Variables

  x + 2 > 3

  For some number x, x + 2 is greater than 3; every number greater than 1 satisfies this, so the set of such x is infinite.

• Variables

  y > x

  For any number x there exists some number y greater than x; the book identifies this assumption as the source of the notion of infinity.

• Variables

  x + y = 1

  A fixed relation between two correlated variables x and y: the pairs (x, y) satisfying it form the aggregate studied as a relation between variables.

• Variables

  y + x = 1

  An equivalent form of the relation x + y = 1, obtained by reordering the terms.

• Variables

  6x + 6y = 6

  An equivalent form of the relation x + y = 1, obtained by multiplying both sides by 6.

• Variables

  y^{2} = x

  A relation between x and y in which x is determined by y as its square; for x = 4, y can be plus or minus 2, so y is not uniquely determined by x.

• Variables

  x + y > 1

  An inequality relating x and y; when either variable is given, an indefinite number of values remain open for the other.

• Variables

  pv = 1

  For a fixed mass of gas at constant temperature, the product of pressure and volume is constant; the book takes the constant to be 1 for illustration.

Problems

No exercises in this chapter.